Enhancing photon correlations through plasmonicstrong couplingR. SÁEZ-BLÁZQUEZ,1 J. FEIST,1 A. I. FERNÁNDEZ-DOMÍNGUEZ,1,3 AND F. J. GARCÍA-VIDAL1,2,4

1Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid,E-28049 Madrid, Spain2Donostia International Physics Center (DIPC), E-20018 Donostia/San Sebastián, Spain3e-mail: a.fernandez-dominguez@uam.es4e-mail: fj.garcia@uam.es

Received 27 June 2017; revised 21 September 2017; accepted 2 October 2017 (Doc. ID 301188); published 1 November 2017

There is an increasing scientific and technological interest in the design and implementation of nanoscale sources ofquantum light. Here, we investigate the quantum statistics of the light scattered from a plasmonic nanocavity coupledto a mesoscopic ensemble of emitters under low coherent pumping. We present an analytical description of theintensity correlations taking place in these systems and unveil the fingerprint of plasmon-exciton-polaritons in them.Our findings reveal that plasmonic cavities are able to retain and enhance excitonic nonlinearities, even when thenumber of emitters is large. This makes plasmonic strong coupling a promising route for generating nonclassicallight beyond the single-emitter level. © 2017 Optical Society of America

OCIS codes: (240.6680) Surface plasmons; (240.5420) Polaritons; (140.6630) Superradiance, superfluorescence; (350.4238)

Nanophotonics and photonic crystals; (030.5290) Photon statistics; (270.0270) Quantum optics.

https://doi.org/10.1364/OPTICA.4.001363

1. INTRODUCTION

Much research attention has focused lately on plasmonic nano-cavities for strong coupling applications. In these devices, theinteraction between surface plasmons (SPs) and quantum emitters(QEs) can be intense enough to yield new hybrid light–matterstates, the so-called plasmon-exciton-polaritons (PEPs) [1].PEPs involving macroscopic QE ensembles have been reportedin planar [2–4] and nanoparticle [5–7] geometries, and they havebeen used for controlling chemical reactions [8,9] or enhancingcharge/energy transport [10,11]. From a purely photonics per-spective, room-temperature PEP lasing has been recently reported[12,13]. However, in order to harness the full potential ofplasmonic cavities for quantum optical applications, plasmonicsystems that display nonlinearities at the single-photon levelwould be highly desirable [14]. This is not possible in macro-scopic ensembles, which present collective boson-like behaviorat pumping levels below the QE saturation regime [15].

Very recently, strong coupling signatures in the power spec-trum of nanogap metallic cavities filled with only a few QEs havebeen reported [16,17]. These experimental advances have beenaccompanied by theoretical efforts aiming to clarify the near-fieldconditions yielding PEPs at the single-emitter level [18].However, the generation of nonclassical light through plasmonicstrong coupling has not been explored yet. In this paper, we fillthis gap by investigating the quantum statistics of the photonsscattered by a nanocavity strongly coupled to a mesoscopic emit-ter ensemble (up to ∼100 QEs) under coherent pumping.

We develop an analytical description of the quantum opticalproperties of the system that allows us to reveal that, contraryto what is expected, plasmonic cavities enhance photon correla-tions in QE ensembles of considerable size under strong couplingconditions.

2. MODEL

Figure 1 depicts the system under study: N identical QEs withtransition dipole moment μQE and frequency ωQE interact withthe near-field ESP (the same for all QEs) of a single SP mode ofenergy ωSP supported by a generic nanocavity. Both subsystemsundergo radiative and nonradiative damping, with decay ratesγQE∕SP � γrQE∕SP � γnrQE∕SP. We consider QEs in which puredephasing is negligible as this process would suppress quantumcorrelations in the emitted photons. Both QEs and SPs arecoherently driven by a laser field EL with frequency ωL. Thesteady-state density matrix ρ for the hybrid system is the solutionof the Liouvillian equation (ℏ � 1)

i�ρ; H � � γSP2

La�ρ� �γrQE

2LS− �ρ� �

γnrQE

2

XNi�1

Lσi �ρ� � 0; (1)

where a, σi, and S− � PNi�1 σi are the annihilation operators

for the SP mode, the ith QE, and the ensemble superradiantstate, respectively. The damping associated with operator O isdescribed by standard Lindblad superoperator LO�ρ� � 2O ρ O†−

fO†O; ρg. Equation (1) reflects that, contrary to nonradiative de-cay, radiation damping is a coherent process that involves only the

2334-2536/17/111363-05 Journal © 2017 Optical Society of America

Research Article Vol. 4, No. 11 / November 2017 / Optica 1363

superradiant state of the QE ensemble (the rest of theensemble states are dark). In the rotating frame, the coherent dy-namics is governed by the time-independent Tavis–CummingsHamiltonian [19]

H � ΔSPa†a� ΔQESz � λ�S�a� S−a†�� ΩSP�a† � a� � ΩQE�S� � S−�; (2)

with ΔQE∕SP � ωQE∕SP − ωL and Sz � 12 �S�; S−�. The QE–SP

coupling is λ � ESP · μQE, while ΩQE � EL · μQE and ΩSP �EL · μSP are the pumping frequencies. Here, μSP is the effectiveSP dipole moment. Once the steady-state density matrix isknown, the first- and second-order correlation functions canbe calculated from the scattered far-field operator at the detectorE−D ∝ μSPa† � μQES

�. Note that we have taken advantage of thesubwavelength dimensions of the system to neglect the differencesbetween the electromagnetic Green’s function describing theemission from the SP and the various QEs in E−

D.

3. RESULTS AND DISCUSSION

Before investigating photon correlations under strong couplingconditions, we consider first both SP and QE subsystems un-coupled. For this purpose, we solve Eq. (1) numerically and com-pute the normalized zero-delay second-order correlation functionin the steady state g �2��0� � hE−

DE−DE

�D E

�Di∕hE−

DE�Di2. This mag-

nitude measures the intensity fluctuations of the emitted light andis related to the probability for two photons to arrive at the sametime at the detector. Values of g �2��0� smaller than 1 indicate anti-bunching, which cannot be achieved with classical light [20]. Weonly consider low laser intensities and study quantum correlationsfar from the pumping regime in which QE saturation becomesrelevant. Figure 2 plots g �2��0� as a function of the laser detuningfor an empty plasmonic cavity (black dashed–dotted line) and en-sembles of different number of emitters (colored solid lines). Forcomparison, the correlation spectra for QEs with γnrQE � 0 are alsoshown (colored dashed lines). The parameters modeling the singleSP mode are ωSP � 3 eV, γSP � 0.1 eV, and μSP � 19 e · nm[5]. Our calculations yield g�2��0� � 1, as expected from theSP inherent bosonic character. For proof-of-principle purposes,we have chosen QE parameters as ωQE � 3 eV, γrQE � 6 μeV(μQE � 1 e · nm), and γnrQE � 15 meV. These values correspondto organic molecules that display very low quantum yield and inwhich collective strong coupling has been already reported [3,13].

As we show below, these types of QEs are also favorable for gen-erating photon correlations. Notice then that for a practical reali-zation of our findings with organic QEs, the experiments shouldbe carried out at low temperature in order to avoid pure dephasingprocesses inside the QEs. For all N , photon statistics are sub-Poissonian (g �2��0� < 1), but the degree of antibunching de-creases rapidly with the ensemble size. As N increases, the systembosonizes and the quantum character of the scattered light is lost[note that g �2��0� � 0.96 for N � 50]. Neglecting nonradiativedamping only leads to an extremely narrow Lorentzian-likeprofile, which suppresses antibunching exactly at zero detuning.This observation is in agreement with the resonance fluorescencephenomenology of a single QE [21], in which no incoherentscattering occurs in the limit of vanishing pumping (saturationeffects in the QE population are negligible). Note that theg �2��0� behavior obtained from our model is in accordance withmore sophisticated descriptions [22] of QE ensembles.

Exact numerical solutions to Eq. (1) can be obtained for strongQE–SP coupling. However, such calculations are only possible forconfigurations involving very small QE ensembles [23], even farfrom the QE saturation regime [24]. In order to circumvent thislimitation and explore photon statistics in mesoscopic ensembles,we map Eq. (1) into the effective non-Hermitian Hamiltonian [25]

H eff � H − iγSP2

a†a − iγnrQE

2Sz − i

γrQE

2S�S−; (3)

where H is given by Eq. (2). Note that H eff depends only on thecollective bright-state operators of the QEs and is independent ofthe dark states of the ensemble (superpositions of QE excitationsthat do not couple to the plasmon or external light), which meansa drastic reduction in the Hilbert space for large N . Equation (3)results from neglecting the refilling terms O ρ O† in the Lindbladsuperoperators in Eq. (1). This approach can be safely employed inthe regime of low pumping, where the ground state can be consid-ered as a reservoir with population equal to 1. In this limit, we cansolve the Schrödinger equation for H eff , treating the coherent driv-ing, EL, as a perturbative parameter [26]. More details on theeffective Hamiltonian approach can be found in Supplement 1.

As we are interested in intensity correlations, we can restrict ourperturbative treatment of Eq. (3) to second order and truncate theHilbert space at two excitations. In the following, for simplicity,

Fig. 1. QE ensemble resonantly coupled to a generic plasmonic cavity.The right inset depicts the two-level QE model.

Fig. 2. Correlation function g�2��0� versus laser detuning for SP (blackdashed–dotted line) and QEs (colored lines) uncoupled. Various ensem-ble sizes are shown, with (solid) and without (dashed) the inclusion ofQE nonradiative decay, γnrQE.

Research Article Vol. 4, No. 11 / November 2017 / Optica 1364

we also assume the plasmonic near-field, ESP, parallel to the laserfield, EL (as, for example, in particle-on-mirror cavities [16]).Moreover, we only consider the optimum configuration for strongcoupling, in which μQE is aligned with ESP. The scattering inten-sity, I � hE−

DE�Di, is given within first-order perturbation theory as

I � �ηNμSPΩSP�2���� ηΔSP � ΔQE∕ηN − 2λ

ΔSPΔQE − N λ2

����2

; (4)

where η � μQE∕μSP � ΩQE∕ΩSP, ΔSP � ΔSP − iγSP∕2, andΔQE � ΔQE − i�γnrQE � N γrQE�∕2. Using second-order perturba-tion theory, the correlation function, g �2��0�, can be expressed as

g �2��0� �����1 − 1

N

�ηΔSP − λ

ηΔSP � ΔQE∕ηN − 2λ

�2 �ΔQE � iN γrQE∕2��ΔQEΔSP � �ΔSP − λ∕η�2 − N λ2��ΔQE � iγrQE∕2��ΔQEΔSP � Δ2

SP − N λ2� − ΔSP�N − 1�λ2����2

: (5)

Note that for γnrQE ≫ γrQE, Eq. (5) yields g�2��0� � �1 − 1∕N �2

at λ � 0 and η → ∞, recovering the flat correlation spectra inFig. 2 for low-quantum-yield QE ensembles.

Figures 3(a)–3(d) render the far-field intensities (top row) andcorrelations (bottom row) for a nanocavity filled with four differ-ent QE ensembles N � 1, 5, 25, and 50, respectively. The hori-zontal and vertical axes correspond to laser frequency and QE–SPcoupling strength, respectively. The latter is expressed through thesingle-emitter cooperativity, C � 2λ2∕γQEγSP, with upper limitC � 2 (λ � 0.03 eV), well below the collective ultrastrong cou-pling regime. We restrict our attention to QE–SP resonant cou-pling and consider the same parameters as in Fig. 2. Although thequantitative results shown in Fig. 3 depend on the specifics of thesystem, we have checked that our findings and their fundamentalimplications remain valid for a wide range of realistic configura-tions. As shown in Supplement 1, the behavior is also very similar

when the SP field is spatially inhomogeneous or when inhomo-geneous broadening is introduced for the QEs (note that theemitters cannot be formally described through a single bright statein these cases but must be treated individually). Dipole–dipoleinteractions among QEs are also analyzed in Supplement 1.Our results reveal that these have a significant impact on photoncorrelations in dense QE ensembles. Interestingly, we find thatantibunching is more robust than bunching when interactionsbecome large.

The complex g�2��0� patterns in Figs. 3(a2)–3(d2) revealthat both photon bunching and antibunching occur in the strong

coupling regime. These panels also show that the main quantumstatistical features emerging at the single-emitter level (which arein qualitative agreement with recent experimental reports onsemiconductor cavities [27,28]) are mostly retained as N in-creases. Up to N ∼ 25, photon emission is antibunched withina narrow frequency window located at C ≲ 1, which implies thatthe single-emitter cooperativity can be considered as the keyparameter determining photon correlations in ensembles contain-ing up to several tens of QEs [29]. Notice also that, as a differencewith high-quantum-yield QEs in low-loss semiconductor cavities,the inherent nonradiative losses of organic molecules and plas-monic systems allow observing antibunching for large C-values(see Supplement 1 for more details). On the other hand, bunchedemission takes place at larger coupling strengths and withinbroader spectral domains for all N . Remarkably, there are spectralwindows in which strong antibunching (g�2��0� ≈ 0) takes place

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

Fig. 3. (a1)–(d1) Scattering intensity I and (a2)–(d2) correlation function g �2��0� versus laser frequency and single emitter cooperativity for various QE–SP systems. In the upper (lower) panels dotted (dashed) lines plot the PEP frequencies (half-frequencies) in the one-excitation (two-excitation) manifold.

Research Article Vol. 4, No. 11 / November 2017 / Optica 1365

even for N � 50, whereas the emission from the uncoupled QEensemble is essentially classical (see Fig. 2). This is the main resultof this paper, namely, that in comparison to the uncoupled sub-systems, collective plasmonic strong coupling can significantlyenhance photon correlations in mesoscopic PEP systems.

By taking advantage of our analytical approach, we can gainphysical insight into the results shown in Fig. 3. The intensitymaps present two scattering maxima, whose origins lie at the de-nominator of Eq. (4). Its vanishing condition yields analyticalexpressions for the dispersion of the lower (LP) and upper (UP)PEPs in the first rung of the Tavis–Cummings ladder. These PEPfrequencies, which naturally incorporate the

ffiffiffiffiffiN

pscaling charac-

teristic of collective strong coupling, are plotted in dotted lines intop panels. Note that the intensity maxima overlap with the PEPdispersion bands, except for N � 1 and C ≲ 0.5. This region,also perceptible for N � 5 at lower C, falls within the weak cou-pling regime, where Fano-like interferences between SP and QEemission gives rise to sharp scattering dips [30]. As N increases,the contrast between UP (brighter) and LP (darker) scatteringpeaks increases. By introducing the PEP frequencies in the nu-merator of Eq. (4), the origin of this asymmetry becomes clear.Neglecting QE and SP damping, we obtain I ∝ �1� ffiffiffiffiffi

Np

η�2,where the upper (lower) sign must be used for LP (UP).Thus, QE and SP dipole moments are antiparallel along theLP dispersion, which diminishes I as N approaches 1∕η2.

In a similar way as in the scattered intensity, we can expect thatthe vanishing of the denominator in the second term of Eq. (5)could give rise to nonclassical light. At N � 1, the resonantfrequencies emerging from this condition are equal to half theenergies of the LP (upper sign) and UP (lower sign) in the secondrung of the Jaynes–Cummings ladder [31]. For N > 1, the samecondition leads to a cubic equation: it accounts for the emergenceof the middle PEPs in the two-excitation manifold (whose realhalf-frequency is equal to ωQE∕SP). Moreover, notice the presenceof the numerator of Eq. (4) in the denominator of the first factorin Eq. (5). As discussed above, this term acquires the form �1 −ffiffiffiffiffiN

pη� at the LP band. Therefore, the darker character of LPs also

makes them more suitable for photon correlations. PEP half-frequencies in the two-excitation manifold are plotted in dashedlines in Figs. 3(a2)–3(d2). The regions of strong photon correlationsdo not occur exactly at one of the polariton energies but slightlyabove the LP dispersion. This indicates that photon correlations,i.e., significant deviations from g �2��0� � 1, do not originate fromtransitions along a single PEP ladder but from the interference inthe emission involving different hybrid states. This underlines thecrucial role that strong coupling plays: while each PEP by itself isquasi-bosonic, the hybridization achieved through strong couplingensures the coexistence of multiple mixed light–matter statesseparated by the Rabi splitting. It is the interference betweenthe emission from these different but closely related states that leadsto strongly nonclassical light emission.

In order to obtain a general view on the degree of bunchingand antibunching attainable through QE–SP coupling, we evalu-ate Eq. (5) at its spectral maxima and minima. Figure 4 explorethese extreme g �2��0� values as functions of cooperativity andnumber of emitters. The inset renders overlapping maps forMax�g �2��0�� (yellow) and Min�g �2��0�� (violet), and the topand bottom panels plot cuts of these maps for various C -values.We can identify three domains according to the statistics of thescattered photons. For small QE ensembles and large C , only

positive correlations take place, as in Figs. 3(a2)–3(c2) forC > 1. In this regime, Max�g �2��0�� grows with increasing cou-pling strength and develops a maximum at N ∼ 10 for all C . Forvery large N , a second domain is apparent. In this limit, PEPsbosonize as the 1∕N factor in Eq. (5) governs g �2��0�, yieldingmaxima and minima approaching 1 monotonically as the numberof QEs increases. Both bunched and antibunched emission takesplace (within different spectral windows) at intermediate N andC . In this third domain, positive correlations decay monotoni-cally with N , whereas negative correlations are enhanced.Min�g �2��0�� diminishes and reaches a minimum value, whichcorresponds to the lowest g �2��0� achievable for a given N andany C (or vice versa). It can be proven that this minimum coin-cides with a sharp dip in the population of the plasmon state(written as a linear combination of PEPs) in the two-excitationmanifold. In the limit of vanishing η (which is a good approxi-mation for our problem at small N ), this condition simplifiesto C � γQE�γSP

2γSP≃ 1

2 . Figure 4 (bottom) shows this minimumdeveloping with increasing cooperativity at N ∼ 10 and reachingg �2��0� � 0 at C � 0.5. Remarkably, this zero in g�2��0� shifts tolarger N for higher cooperativity, yielding strong photon anti-bunching at ensemble sizes as large as 100 QEs. Therefore, asanticipated in Fig. 3(d2), plasmonic strong coupling leads tothe emergence of quantum nonlinearities in large excitonic sys-tems, which would present g �2��0� ≃ 1 when not coupled tothe plasmonic nanocavity.

4. CONCLUSION

We have investigated the complex photon statistics phenomenol-ogy that emerges from the strong coupling of a mesoscopicensemble of quantum emitters and a single plasmon mode sup-ported by a generic nanocavity. We have presented an analyticalmethod describing the optical response of these systems underlow-intensity coherent illumination. Our approach provides in-sights into the role that both the plasmon-exciton-polariton lad-der and its tuning through the single emitter cooperativity play inthe emission of strongly correlated (bunched and/or antibunched)light. Finally, our results demonstrate the robustness of thesecompound systems against bosonization effects, predicting strong

Fig. 4. Maximum (top) and minimum (bottom) correlation functionsas functions of the QE ensemble size for several values of the single emittercooperativity. The inset in the upper panel shows the map of photon pos-itive (yellow) and negative (violet) correlations as functions of N and C .

Research Article Vol. 4, No. 11 / November 2017 / Optica 1366

intensity correlations at considerable ensemble sizes. Our theoreti-cal findings demonstrate the feasibility and establish experimentalguidelines toward the realization of nanoscale nonclassical lightsources operating beyond the single-emitter level.

Funding. FP7 Ideas: European Research Council (IDEAS-ERC) (ERC-2011-AdG 290981, ERC-2016-STG-714870,FP7-PEOPLE-2013-CIG-618229, FP7-PEOPLE-2013-CIG-630996); Ministerio de Economía y Competitividad(MINECO) (FIS2015-64951-R, MAT2014-53432-C5-5-R,MDM-2014-0377).

See Supplement 1 for supporting content.

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